Gallai-Ramsey numbers for rainbow P5 and monochromatic fans or wheels

Given two graphs G,H and a positive integer k, the Gallai-Ramsey number grk(G:H) is the minimum integer N such that for all n≥N, every exact k-edge-coloring of Kn contains either a rainbow copy of G or a monochromatic copy of H. A fan Fq is obtained from a matching of size q by adding a vertex adjacent to all vertices in the matching; while a wheel graph Ws is obtained from a cycle of size s−1 by adding a vertex adjacent to all vertices on the cycle. In this paper, we determine either the exact values or some bounds for the Gallai-Ramsey numbers grk(P5:H)(k≥3) where H is either a fan or a wheel graph.